Laminated conductor



Oct. 30, 1956 H 5, BLACK ET AL 2,769,150

LAMINATED CONDUCTOR Filed NOV. 14, 1952 A 2 Sheets-Sheet l VEA/Tops H. 5. BLACK -N 5. P MORGAN JR.

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ATTORNEY Oct. 30, 1956 Filed Nov"Y 14, 1952 2 Sheets-Sheet 2 ATTO/QNEV LAMINATED CONDUCTOR Harold S. Black, New Providence, and Samuel P. Morgan, Jr., Morristown, N. J., assiguors to Bell Telephone Laboratories, Incorporated, New York, N. Y., a corporation of New York Application November 14, 1952, Serial No. 320,468

4' Claims. (Cl. 3133-96) This invention relates to electrical conductors and more particularly to composite conductors formed of a multiplicity of insulating conducting portions and which have come to be known as Clogston conductors. Specifically, the invention relates to a particular type of Clogston conductor which comprises a single coaxial stack of insulated metallic cylinders and which is called a Clogston 2 conductor.

It is an object of this invention to improve the current distribution in Clogston 2 conductors, and particularly to effect such improvement by the proportioning of the insulating and metallic material in the stack.

`In a copending application of A. M. Clogston, Serial No. 214,393, filed March 7, 1951, there are disclosed a number of composite conductors each of which comprises a multiplicity of insulated conducting elements of such number, dimensions, and disposition relative to each other and to the orientation of the electromagnetic wave being propagated therein as to effect a more favorable distribution of current and field within the conducting material. In one specific embodiment shown in Fig. 17A of the Clogston application, a coaxial transmission line has the entire region between an outer sheath and an inner core filled with alternate thin cylinders or laminae of metal and dielectric material, respectively. The metal cylinders are made as thin as possible compared with the skin depth where is given by the expression where is expressed in meters, is the frequency in cycles per second, u is the permeability of the metal in henries per meter, and o' is the conductivity of the metal in mhos per meter. The factor measures the distance in which the current or field penetrating into a slab of the material many times in thickness will decrease by one neper, i. e., their amplitude will become equal to 1/e=0.3679 times their amplitude at the surface of the slab.

This factor called one skin thickness or one skin depth, is many times, for example, 10, 100, or even 1000 times larger than the thickness of each metal or insulating lamina. It has been found that when the composite conductor has such a laminated structure, it will propagate a wave at a certain critical velocity which is determined by the geometry of the structure and this wave` will penetratefurther into the conductor and under proper conditions will have lower attenuation than the transmission mode of a conventional coaxial cable of the same size. `The critical velocity mentioned above is determined by `the thickness of the metal and insulating laminae, the dielectric constant of the insulating laminae and the permeabilities of the conducting and insulating laminae. It is further shown in the aforementioned Clogston application that the critical velocity is achieved `by equating the product of the average dielectric constant of the laminated stack and the average ICC permeability thereof to the product of the dielectric constant and permeability of the remaining portion of the cable through which the wave propagates. When these products are equal, the velocity of propagation of the waves is substantially uniform throughout the crosssectional area of the cable.

It has been discovered that in Clogston 2 conductors of the type just described, the attenuation constant increases gradually frorn very low frequencies up to a first certain frequency fr', in practice a few kilocycles, and is then substantially uniform up to a second certain frequency f2', which may in practice be many megacycles. The values of these two certain frequencies f1 and f2' are dependent on various factors among which are the composition, thickness, and proportioning of the various laminae in the composite conductors. Above this certain second frequency f2' the attenuation constant rises, in many cases quite steeply, to a third certain frequency f3', which may be hundreds of times f2', at which frequency f3 the rate of increase of attenuation constant decreases somewhat. ln the description which follows, all frequencies above f2 are called elevated frequencies. The present invention is concerned with the proportioning of Clogston 2 conductors to give optimum performance in this elevated frequency area.

In the above-identified Clogston application, is is pointed out that for certain conditions the thickness W of each metal lamina should be twice the thickness t of each insulating lamina in the stack. In other Words, 0, the fraction of the stack filled with conducting material is equal to 2/3. This 022/3 relationship is particularlyapplicable in cables having substantially infinitely thin insulating and metal layers and for frequencies below f2. However, for finite thicknesses of metal and insulating laminae and especially for frequencies in the eleva-ted range, that is, above f2', it has been discovered that other values of H produce lower attenuation.

It is accordingly another object of this invention to produce as low a loss as possible at elevated frequencies in Clogston 2 cables having conducting and insulating layers of finite thicknesses.

The present invention is based on the discovery that the attenuation can be minimized in the elevated frequency region, that is, for frequencies above f2', for various assigned conditions by fixing the value of 6 at something other than this previously prescribed value, that is, something other than 2/3, the specific value of 0 chosen being dependent upon the particular conditions. For example, it is possible to select a value of 0 so that the attenuation of the cable is minimized for a particular frequency. Also, it is possible to picka value of 0 so that the attenuation of the cable is maintained below a predetermined value for the widest possible frequency range.

The invention will be more readily understood by referring to the following description taken in connection with the accompanying drawings forming a part thereof in` -and 2 show, by way of example for purposes ofrillustration, a conductor 10 in accordance with the invention, Fig. 1 being an end View and Fig. 2 being'a longitudinal View. The conductor comprises a central core 11 which may be either of metal or of dielectric material, an outer sheath 12 of metal or other suitable shielding material, and a composite conductor or stack 13 filling the entire space between the members Sill and 12 and comprising many laminations of metal i4 spaced by insulating material 1S. As disclosed in the abovementioned Clogston application, with particular reference to Fig. 17A which shows a composite conductor similar to the composite conductor l0, each of the metal layers 14 is made very thin compared to the skin depth of the conductor being used, which for example, can be copper, silver, or aluminum. The insulating layers 15 are also made very thin and may be of any suitable material. Examples of satisfactory insulating materials are: polyethylene, polystyrene quartz, .and polyfoam. The stack 13 has, for example, ten or a hundred or more metal and insulating layers 114 and l5 and since there are a large number of insulating and metallic layers, it makes no difference whether the first or the last layer of the stack 13 is of metal or insulation.

Fig. 3 shows a curve, drawn on a log-log scale, of attenuation constant a versus frequency for a particular cable of the general type shown in Figs. 1 and 2. It will be appreciated that this curve is merely representative and that a different curve can be drawn for a Clogston 2 conductor of different size and proportioning. Hence exact values taken from this curve are of no significance. In general, the shapes of all curves of this kind, however, are similar to one another. This curve can be considered as being in four parts. The low frequency region A extends from very low frequencies up to a frequency designated in Fig. 3 as f1', a second region B extends from frequency fr to a frequency designated f2', a third region C extends from f2 to a higher frequency f3', and a fourth region D cornprises all frequencies above fs. The frequencies of points f1', fz and f3 are determined by the points at which the dotted line curve (approximate curve shown in Fig. 3) changes slope. Referring now to this approximate dotted curve in Fig. 3, it will be noted that in the region A, the attenuation constant increases substantially in proportion to the square root of the frequency to approximately the frequency of fr', which in practice may be of the order of a few kilocycles. Between f1' and f2 the attenuation constant is substantially independent of frequency and f2 may be as high as a few megacycles. Between f2 and f3' the attenuation constant is substantially proportional to the square of the frequency, While in the region D above f3', which may be of the order of a hundred megacycles, the attenuation constant is substantially constant to the square root of the frequency. As pointed out above, the frequencies above f2' are designated elevated frequencies, and it is within this region that the improvement in attenuation constant is most marked. Prior to the present invention, the emphasis on minimizing the attenuation constant has been primarily in the region below fz.

In accordance with the present invention the value of 6 is selected to produce desirable operating characteristics for various assigned conditions in the elevated frequency area. Before specifying these conditions it seems desirable to present a mathematical analysis of the cable 10 shown in Figs. 1 and 2 in order to provide a proper background for the discussion of the present invention.

In this analysis:

a=inner radius of stack b=outer radius of stack t1=thickness of each conducting layer (called W in the above-identified Clogston application) tz=thickness of each insulating layer (called t in the Clogston application) 0:11/ (ll-l-tz); fraction of stack lled with conducting material 0m=optirnum value of 0 as defined herein 2T1=6(b-a); total thickness of conducting material in stack e2=dielectric constant of the insulating layers =e2/(l-0); average dielectric constant of stack itl=permeability of conducting layers tt2=permeability of insulating layers ,tzul-l-(l-Miq; average permeability of stack (riz-conductivity of conducting layers 5:7961; average conductivity of stack lzskin depth in conducting layers tt=attenuation constant a0=lowfrequency (at) attenuation constant of any Clogston 2 conductor 00:minimum value of a0 (obtained with 0:2/3 for non-magnetic lines) f=frequency fa=a characteristic frequency associated with a Clogston 2 cable and defined by Equation 9 below fm=highest frequency in the operating band X=a root of Equation 3 below p=a mode number; the principal mode correspondsto fp(tz/b)=a numerical function; for p=l, f1(a/b) is plotted against a/ b in Fig. 4. The attenuation constant of the pth mode in a coaxial Clogston 2 cable is given approximately by To a rough approximation Xp is equal to p1r/(b-a) accurate Values can be found from curves or tables. In general one can write Panda/b) Xzf"1 b a where fp(a/b) is a numerical factor of the order of unity. For the principal mode (p=1), f1(a/ b) is plotted against a/ b in Fig. 4. Equation 2 for the attenuation constant is valid from frequencies where 51-T1(or T 1/p for the higher modes) to frequencies where ltl, so that it may be applied over the `Whole range of engineering interest.

The problem of minimizing the attenuation constant of a Clogston cable of given size, made of given materials, at a preassigned top frequency fm, by proper choice of the value of 0 will rst be considered. For deniteness it can be supposed that the thickness t1 of the conducting layers is given, and that 0 is varied by varying the thickness of the insulating layers.

The attenuation constant am of the cable at the frequency fm is given by d w/itxwfeaarwmfma 7l-@Wouw(Morrill/20m By logarithmic differentiation it is found that the righthand side of (4) is a minimum when 0 satisfies the expression t t materials used are non-magnetic, the formal results simplify a good deal. Equation 5 then becomes duction of the characteristic attenuation constant am and the characteristic frequency fc defined as follows:

gl. *|:l +912 g3] iou 31/:3-0l1m1/2 A 4 "faz The relation satised by the optimum value of 0, namely Equation 6, may be reduced when,pi,=u2 to the cubic equation The root 0m of Equation 11 is plotted against the ratio fin/fc in Fig. 5. It is worth noting that 6m approaches 2% as fm/fc approaches zero, and 0m is approximately equal to Zic/3fm when fm fc. The value of atm/ot00 corresponding to 0m is given by -Lil ,SM fr 00 :Nami-0.01m 4 fg and is plotted against fin/fc in Fig. 6. The ratio xm/am approaches unity as ,fm/fc approaches zero, and is approximately equal to 2fm/ 3 fc when fm fc.

Thus, given a particular top frequency fm, fc can be calculated from Equation 9, with reference also to Fig. 4 for the principal mode or to a similar curve for other modes. One then calculates the value of the ratio fin/fc and reads oif the value of 0m corresponding to this ratio from the curve of Fig. 5

A closely related problem will now be considered. Given a certain maximum attenuation constant am, the value of 0 to maximize the frequency band over which the attenuation constant of the cable does not exceed the preassigned value am will be determined.

In the general case Where the permeabilities a, and a, of the conducting and insulating laminae may be different, the frequency fm at which a is equal to am is given by It is assumed throughout that am is at least as large as the minimum attenuation which can be achieved with this cable, so that fm as given by Equation 13 is real.

It is desired to maximize fm considered as a function of 0, that is, to choose the conductor-to-insulator ratio, which is equal to 0/(1-0), so as to maximize the frequency band over which a does not exceed am. Differentiation shows that 0 must satisfy the relation If both sides are squared and cleared of fractions, Equation 14 reduces to a quartic equation in 0, which may be solved either by formulas or by successive approxirnations in any particular case. Denoting by 0m the root of Equation 14 which lies between zero and one, the maxi- 6 mum value of fm is found by substituting 0m for 0 in Equation 13.

If the conducting and insulating layers have equal permeabilities (tif-nq), then Equation 5 becomes Lg 3\/(1-e)1/2 t 15) fc 3 (100 02 in terms of the characteristic attenuation constant am and the characteristic frequency fc defined by Equations 8 and 9 respectively. The equation satisfied. by the optimum value of 6' takes the form (2H-02) 8 in@ 1/1-0 34 am and if 6m is the root of this equation between zero and one, the maximum value of fm is given by f"`30m 2-0m The ratio xm/a0, is just the ratio of the maximum permissible attenuation constant to the minimum low-frequency attenuation constant attainable in a Clogston cable of the same diameter. The valueof 0m obtained from Equation 16 is plotted against im/am in Fig. 7; 0m is equal to 2/3 when atm/a0, is equal to unity, and for large values of the attenuation ratio 0mz4a0/3\/3am. The relationship between fm/fc and vtm/am, for the optimum cable is the same as that shown in Fig. 6. For fat/0021, fm/fc=0 to the present approximation; and for large values of 0cm/Iam, fm/ fc\/3 0am/ 21x00.

Thus, given a particular value of am, aco can be calculated from Equation 8, with reference `also to Fig. 4 for the principal mode or to a similar curve for other modes. One then calculates the value of the ratio ccm/am, and reads oi the value of 6m corresponding to this ratio from the curves of Fig. 7.

It is obvious that in a (non-magnetic) Clogston cable with 6%#2/3, the low-frequency attenuation constant au will be greater than 00. In fact, 1/2am am, which means that in choosing the optimum value of 0 as has been done above, not only has the top frequency fm been increased but also the attenuation vs. frequency characteristic is flattened by raising the low-frequency end. This result reduces the problem of equalizing the Clogston cable over an extended frequency band.

It is to be understood that the above-described embodiments are illustrative of the application of the principles of the invention. Numerous modifications may be devised by those skilled in the art without departing from the spirit yand scope of the invention.

What is claimed is:

l. A cable for the propagation of electromagnetic Waves in the elevated frequency range comprising a coaxial stack of conducting members spaced by insulating members, there being a suicient number of conducting members to carry a substantial portion of the current, the thickness of each conducting member being less than a skin depth at the highest frequency of operation of said cable, the thickness of each conducting member being zi, the thickness of each insulating member being t2, the ratio 0 of ri to (t1-|42) having a value between zero and one and satisfying the equation where a, is the permeability of the conducting members, ,u2 is the permeability of the insulating layers, a, is the conductivity of the conducting members, fm is the maximum frequency of operation of the cable, and XD is the pth root of the Bessel equations where a is the inner radius of the stack and b is the outer radius of the stack, and p is equal to the mode order of the Waves being propagated, the permeability and dielectric constant of the insulating members and the permeability vof the conducting members being such that the velocity of propagation is substantially uniform throughout the cross-sectional area of said stack.

2. A cable as claimed in claim 1 wherein 9 is approximately equal to where where am is the maximum allowable attenuation, a, is the conductivity of the conducting members, e2 is the dielectric constantvof the insulating members, ,u1 is the permeability of the conducting members, a2 is the permeability of the insulating members, and Xn is the pth root of the Bessel equations J1(xa)N1(xb)J1(xb)N1(xa)=0 Y where a is the inner radius of the stack and b is the outer radius of the stack, and p is equal to the mode order of the waves being propagated, the permeability and dielectric constant of the insulating members and the permeability of the conducting members being such that the velocity of propagation is substantially uniform throughout the cross-sectional area of said stack.

4. A cable as claimed in claim 3 wherein 0 is approximately equal to 41100 Qn/gam where ,100:37 8x12 4 eL-U;

References Cited in the file of this patent UNITED STATES PATENTS 1,701,278 Silbermann Feb. 5, 1929 2,433,181 White Dec. 23, 1947 

